Multi-piece multi-dimensional functionally graded material optimization method, device, equipment and medium
By constructing a mesh model of multiple three-dimensional functionally graded materials and using PHT-AIGA for adaptive solution and iterative refinement, the problems of insufficient computational efficiency and accuracy in traditional methods are solved, and efficient multi-objective optimization design is achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-16
- Publication Date
- 2026-03-27
AI Technical Summary
Traditional methods for analyzing and optimizing multidimensional functional graded materials (FJDs) are insufficient in terms of computational efficiency and accuracy, especially in the numerical calculation of complex multi-layer triaxial FJDs, where they are difficult to meet the requirements for computational accuracy and efficiency.
Multi-layer three-dimensional functional graded material porous structures are constructed using B-spline or NURBS-based modeling and design meshes. Adaptive solution and iterative refinement are performed using PHT-AIGA. Design variables are optimized through geometric analysis based on PHT spline basis, superconvergent recovery basis error estimation algorithm, and Doveer labeling strategy to achieve multi-objective optimization.
It significantly improves the computational accuracy and optimization efficiency of multi-sheet three-dimensional functionally graded materials, reduces computational costs, and shortens the design cycle.
Smart Images
Figure CN121744649A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of functionally graded materials technology, and in particular to a method, apparatus, equipment and medium for optimizing multi-sheet multidimensional functionally graded materials. Background Technology
[0002] Functionally graded materials (FGMs) are a new type of composite material composed of two or more different components. Their characteristic is that the material properties change continuously along the spatial position. Traditional one-dimensional functionally graded materials (1D FGMs) mostly achieve property changes along the thickness direction. However, their applicability is limited and it is difficult to meet the multi-dimensional gradient requirements under complex service conditions. Therefore, the development of two-dimensional (2D) or three-dimensional (3D) multi-dimensional functionally graded materials (MFGMs) is of great engineering significance.
[0003] In terms of numerical analysis, the finite element method (FEM) is the traditional main method for MFGM analysis and optimization. However, this method relies on mesh generation, which has problems such as low computational efficiency and large geometric errors. Isogeometric analysis (IGA) is adopted, which uses spline basis functions such as B-splines and non-uniform rational B-splines (NURBS) to achieve consistency between CAD models and numerical analysis models, greatly improving computational accuracy and efficiency, and has become the mainstream method for FGM analysis.
[0004] However, when dealing with complex multi-sheet triaxial functional graded materials, the B-splines or NURBS tensor product structures used in traditional IGA are rigid, lacking local refinement capabilities and unable to perform reasonable mesh generation, resulting in low computational efficiency and difficulty in meeting computational accuracy requirements. This makes it difficult to handle numerical calculations of complex structures and limits the multi-objective optimization design of functional graded materials. Summary of the Invention
[0005] In view of this, it is necessary to provide a method, apparatus, device and medium for optimizing multi-sheet multidimensional functional graded materials, so as to solve the technical problem of low accuracy and efficiency in multi-objective optimization of complex multi-sheet triaxial functional graded materials.
[0006] To address the aforementioned problems, in a first aspect, the present invention provides a method for optimizing multi-sheet multidimensional functionally graded materials, comprising: Modeling meshes and design meshes are established based on B-splines or NURBS respectively to construct multi-sheet three-dimensional functional graded material porous structures. The multi-sheet three-dimensional functional graded material porous structures include multiple control points and design variables defined at the control points. The design variables include material volume fraction and porosity. The constructed PHT-AIGA is used to adaptively solve and iteratively refine the geometric configuration of porous structures of multi-layer three-dimensional functional graded materials to obtain a numerical analysis template mesh that meets the error threshold. The PHT-AIGA includes geometric analysis such as PHT spline basis, superconvergent recovery basis error estimation algorithm and Doveer marking strategy. A multi-objective optimization function is constructed with the objectives of minimizing mass and maximizing the fundamental frequency of free vibration. The multi-objective optimization function is iteratively solved to obtain the optimized design variables. The design variables at the control points of the design grid are used as the optimization basis. The population fitness during the multi-objective optimization process is evaluated on the numerical analysis template grid using the PHT-AIGA.
[0007] In one possible implementation, the three-dimensional functionally graded material is composed of metallic and ceramic materials; the construction of a multi-layered three-dimensional functionally graded material porous structure based on B-splines or NURBS to establish a modeling mesh and a design mesh respectively includes: Modeling meshes are built based on B-splines or NURBS to construct the geometric configurations of multiple three-dimensional functionally graded materials; A design mesh is established based on B-splines or NURBS to construct a monolithic hexahedral design domain, wherein the monolithic hexahedral design domain completely covers the geometric configuration; Interpolate the design variables at the control points in the design grid based on B-splines or NURBS to obtain the value of the design variables at any point in the hexahedral design domain; By mapping, the values of design variables in the hexahedral design domain established by the design mesh are transferred to the geometric configuration positions established by the modeling mesh under the corresponding coordinates, so as to construct a multi-layer three-dimensional functional graded material porous structure.
[0008] In one possible implementation, the step of adaptively solving and iteratively refining the geometry of a multi-layer three-dimensional functionally graded material porous structure using the constructed PHT-AIGA to obtain a numerical analysis template mesh that meets the error threshold includes: Step 1: Establish the elastic dynamics governing equations for porous structures of multi-layer three-dimensional functionally graded materials based on three-dimensional elasticity theory and Nitsche's method; Step 2: Use the modeling mesh as the initial analysis mesh; Step 3: Based on the aforementioned elastic dynamics control equations, the geometric configuration is numerically solved using geometric analysis methods such as PHT spline basis according to the analysis mesh to obtain the restored stress field; Step 4: Use the superconvergent recovery basis error estimation algorithm to estimate the error of the original stress field and the recovered stress field, obtain the global relative error, and determine the subdomain with the maximum global relative error in the geometric configuration of the multi-piece. Step 5: Use the Doffler labeling strategy to label the subdomain of the maximum error estimate; Step 6: Based on the PHT spline local subdivision strategy, subdivide the marked subdomains into a mesh to obtain the updated analysis mesh; Step 7: Repeat steps 3 to 6 until the global relative error is less than or equal to a preset threshold, then stop the iteration to obtain a numerical analysis template mesh that meets the error threshold.
[0009] In one possible implementation, the elastic dynamics governing equations are: , , , , , , , in, For the first three-dimensional functionally graded porous structure of multiple sheets Individual pieces, For stress tensor, For volume forces, For density, For displacement, For subfield symbols, For Dirichlet boundaries, For Neumann boundary, For gradient operators, The boundary displacement of the subdomain on the Dirichlet boundary. The traction force specified by the Neumann boundary for the subdomain. This is the initial displacement. The initial velocity, For the internal boundaries of the two sheets, Let be the normal vector of the first piece of material from its internal boundary outwards. Let be the normal vector of the second body from its inner boundary outwards. Let the stress tensor of the first sheet be... For the stress tensor of the second sheet, For the displacement of the first piece, For the displacement of the second piece, For the first The initial velocity of the sheet.
[0010] In one possible implementation, a multi-objective optimization function is constructed with the optimization objectives of minimizing mass and maximizing the fundamental frequency of free vibration. The multi-objective optimization function is then iteratively solved to obtain the optimized design variables, including: An initial population of NSGA-III is constructed based on the design variables at the control points of the design grid, where each individual in the population has a unique material gradient distribution and porosity gradient distribution. The fitness of individuals in the initial population is evaluated using PHT-AIGA on the numerical analysis template grid. During each evaluation, superconvergent recovery basis error estimation is performed. When the global relative error is less than a set threshold, the calculation result is saved as the fitness of the individual in the population. When the global relative error is greater than the set threshold, the numerical analysis template grid is further refined until the global relative error is less than the threshold. The result is saved as the fitness of the current individual in the population. Offspring are generated through crossover and mutation operations using NSGA-III. Combining non-dominated sorting and a reference point-based selection mechanism, the system iteratively approximates the final Pareto front, and optimized design variables are obtained based on the Pareto front.
[0011] In one possible implementation, the multi-objective optimization function is: , in, For designing the grid Material volume fraction at each control point For designing the grid Porosity at each control point For modeling the mesh Plate thickness at each control point It is the volume fraction of the material. The elastic modulus of the material, The mass density of the material, For dimensionless frequency, The characteristic length of the geometric configuration, For the initial uniform thickness, It is a geometric configuration structural domain.
[0012] In one possible implementation, the constraints of the multi-objective optimization function are: , in, The total integral of the material. The total porosity of the material.
[0013] In a second aspect, the present invention also provides a multi-sheet multidimensional functionally graded material optimization device, used for: A porous structure construction module is used to establish a modeling mesh and a design mesh based on B-splines or NURBS, respectively, to construct a multi-sheet three-dimensional functional graded material porous structure. The multi-sheet three-dimensional functional graded material porous structure includes multiple control points and design variables defined at the control points. The design variables include material volume fraction and porosity. The template mesh acquisition module is used to adaptively solve and iteratively refine the geometric configuration of the porous structure of multiple three-dimensional functional graded materials using the constructed PHT-AIGA to obtain a numerical analysis template mesh that meets the error threshold. The PHT-AIGA includes geometric analysis such as PHT spline basis, superconvergent recovery basis error estimation algorithm and Doveer marking strategy. The optimization module is used to construct a multi-objective optimization function with the objectives of minimizing mass and maximizing the fundamental frequency of free vibration. The multi-objective optimization function is iteratively solved to obtain the optimized design variables. The design variables at the control points of the design grid are used as the optimization basis, and the population fitness during the multi-objective optimization process is evaluated on the numerical analysis template grid using the PHT-AIGA.
[0014] Thirdly, the present invention also provides an electronic device, comprising: a processor and a memory; The memory stores a computer-readable program that can be executed by the processor; When the processor executes the computer-readable program, it implements the steps in the multi-sheet multidimensional functionally graded material optimization method described above.
[0015] Fourthly, the present invention also provides a computer-readable storage medium for storing a computer-readable program or instructions, which, when executed by a processor, can implement the steps in the multi-sheet multidimensional functionally graded material optimization method described in any one of the above-mentioned method items.
[0016] The beneficial effects of this invention are as follows: A multi-layered three-dimensional functionally graded material (FJT) porous structure is constructed based on B-splines or NURBS. Multiple layers are modeled using B-splines or NURBS to construct complex geometric configurations. The constructed PHT-AIGA is used to adaptively solve and iteratively refine the geometric configurations of the multi-layered FJT porous structure, obtaining a numerical analysis template mesh that meets the error threshold. PHT-AIGA includes PHT spline-based isogeometric analysis, a superconvergent recovery basis error estimation algorithm, and a Doffler marking strategy. A multi-objective optimization function is constructed with the minimum mass and maximum free vibration fundamental frequency as optimization objectives. The multi-objective optimization function is iteratively solved to obtain optimized design variables. The design variables at the control points of the design mesh are used as the optimization basis. The PHT spline-based adaptive isogeometric analysis method can intelligently refine the mesh, achieving higher computational accuracy with significantly fewer degrees of freedom. When applied to fitness evaluation in multi-objective optimization algorithms, it ensures the accuracy of the analysis, greatly reduces computational costs, effectively shortens the optimization design cycle, and improves the accuracy and efficiency of multi-objective optimization of multi-layered three-dimensional FJTs. Attached Figure Description
[0017] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.
[0018] Figure 1 A flowchart illustrating an embodiment of the optimization method for multi-layer multidimensional functionally graded materials provided by the present invention; Figure 2 A schematic diagram of mesh generation for the multi-sheet multidimensional functionally graded material optimization method provided by the present invention; Figure 3 A schematic diagram of the design mesh for the multi-sheet multidimensional functionally graded material optimization method provided by this invention; Figure 4 The Pareto front results for the optimization method of multi-sheet multidimensional functionally graded materials provided in this invention are shown in the figure. Figure 5 The result is a diagram showing the material distribution of a representative elite solution of the multi-sheet multidimensional functionally graded material optimization method provided by this invention. Figure 6 The porosity distribution of a representative elite solution of the optimization method for multi-sheet multidimensional functionally graded materials provided by this invention is shown in the figure. Figure 7 A schematic diagram of a structure of an embodiment of the multi-sheet multidimensional functionally graded material optimization device provided by the present invention; Figure 8A schematic diagram of an embodiment of the electronic device provided by the present invention. Detailed Implementation
[0019] Preferred embodiments of the present invention will now be described in detail with reference to the accompanying drawings, which form part of this application and are used together with the embodiments of the present invention to illustrate the principles of the present invention, but are not intended to limit the scope of the present invention.
[0020] In this document, the term "embodiment" means that a specific feature, structure, or characteristic described in connection with an embodiment may be included in at least one embodiment of the invention. The appearance of this phrase in various places throughout the specification does not necessarily refer to the same embodiment, nor is it a separate or alternative embodiment mutually exclusive with other embodiments. It will be explicitly and implicitly understood by those skilled in the art that the embodiments described herein can be combined with other embodiments.
[0021] This invention discloses a method, apparatus, device, and medium for optimizing multi-sheet multidimensional functionally graded materials, which can be used in a computer. The method, apparatus, or computer-readable storage medium involved in this invention can be integrated with the aforementioned apparatus or be relatively independent.
[0022] One specific embodiment of the present invention discloses a method for optimizing multi-sheet multidimensional functionally graded materials, which can be executed by a computer, specifically by one or more processors of the computer. For example... Figure 1 As shown, the optimization method for multi-sheet multidimensional functionally graded materials includes: S101. Establish modeling mesh and design mesh based on B-spline or NURBS respectively to construct a multi-layer three-dimensional functional graded material porous structure. The multi-layer three-dimensional functional graded material porous structure includes multiple control points and design variables defined at the control points. The design variables include material volume fraction and porosity. It should be noted that using B-splines or NURBS for multi-piece modeling and constructing complex geometric configurations ensures the accuracy and reliability of numerical analysis of complex combined structures.
[0023] S102. The constructed PHT-AIGA is used to adaptively solve and iteratively refine the geometric configuration of the porous structure of multiple three-dimensional functional graded materials to obtain a numerical analysis template mesh that meets the error threshold. PHT-AIGA includes geometric analysis such as PHT spline basis, superconvergent recovery basis error estimation algorithm and Doveer marking strategy. It should be noted that the adaptive isogeometric analysis method based on PHT splines can intelligently refine the mesh, achieving higher computational accuracy with significantly fewer degrees of freedom.
[0024] S103. Construct a multi-objective optimization function with the minimum mass and maximum free vibration fundamental frequency as optimization objectives. Iterate the multi-objective optimization function to obtain the optimized design variables. The design variables at the control points of the design grid are used as the optimization basis. PHT-AIGA is used on the numerical analysis template grid to evaluate the population fitness during the multi-objective optimization process. It should be noted that by incorporating adaptive geometric analysis methods based on PHT splines into the fitness evaluation of multi-objective optimization algorithms, the accuracy of the analysis is ensured, the computational cost is greatly reduced, and the optimization design cycle is effectively shortened.
[0025] In some embodiments, in step S101, a modeling mesh and a design mesh are established based on B-spline basis functions or NURBS to construct a multi-layer three-dimensional functionally graded material porous structure. The multi-layer three-dimensional functionally graded material porous structure includes multiple control points and design variables defined at the control points. The design variables include material volume fraction and porosity. A modeling mesh is established based on B-splines or NURBS to construct the geometric configuration of the multi-layer three-dimensional functionally graded material, defining the multi-layer three-dimensional functionally graded material as a three-dimensional elastic body of arbitrary shape. It consists of multiple subdomains, with two non-overlapping subdomains. and From internal boundary Separate, and Representing subfields respectively and from The outward normal vector, the structural domain of the multi-layer 3D functional graded material contains many pores, which can be considered to be uniformly or non-uniformly distributed. The multi-layer 3D functional graded material structure is made of ceramics and metals, and the materials that can be selected include zirconium oxide, alumina, SUS304, aluminum, etc., with their volume fractions being respectively... and And it changes smoothly within the structural domain, and satisfies The conditions define the junction vectors of B-splines or NURBS. ,in, For B-splines or NURBS, the order is... To determine the number of control points and basis functions, the geometric configurations of multiple 3D functionally graded materials are constructed using B-splines or NURBS junction vectors and control points. The B-spline basis functions are defined using the Cox-de-Boor recursive formula. , , Based on the B-spline basis function, weight values are introduced. To obtain NURBS basis functions: , Two-dimensional and three-dimensional NURBS basis functions are constructed using tensor products of one-dimensional NURBS basis functions. The two-dimensional and three-dimensional NURBS basis functions are as follows: , , The modeling mesh of the parameter space is constructed by dividing the knot vectors and distributing control points using B-splines or NURBS. Each non-zero interval of the knot vector corresponds to a parameter element. The geometric configuration in the physical space is established by defining the values of the control points. A design mesh is established based on B-splines or NURBS to construct a monolithic hexahedral design domain. This domain should completely cover the geometric configuration. Design variables at control points within the mesh are interpolated using B-splines or NURBS to obtain the design variable values at any point within the hexahedral design domain. The material volume fraction and porosity values are as follows: , , , in, For the first Material volume fraction at each control point For the first Porosity at each control point For the corresponding B-spline or NURBS basis functions, This represents the total number of control points in the design grid. By obtaining equivalent material properties through an improved hybrid rule, the structural properties of the material can be obtained, which can be expressed as: , in, For effective material properties, elastic modulus Poisson's ratio and density , and These are the material properties of ceramics and metals, respectively. For porosity, when When constant, the porosity distribution is uniform; By mapping, the values of design variables in the hexahedral design domain established by the design mesh are transferred to the geometric configuration positions established by the modeling mesh under the corresponding coordinates, so as to construct multiple three-dimensional functional graded material porous structures. That is, under the corresponding physical space coordinates, the corresponding material volume fraction, porosity and other design variable information are assigned, thereby establishing multiple 3D-FGM porous structures.
[0026] In some embodiments, in step S102, the constructed PHT-AIGA is used to adaptively solve and iteratively refine the geometric configuration of the porous structure of multiple three-dimensional functionally graded materials to obtain a numerical analysis template mesh that meets the error threshold. PHT-AIGA includes geometric analysis such as PHT spline basis, a superconvergent recovery basis error estimation algorithm, and a Doffler marking strategy. The specific steps are as follows: Step 1: Establish the elastic dynamics governing equations for three-dimensional functionally graded porous structures based on three-dimensional elasticity theory and Nitsche's method: , , , , , , , in, For the first three-dimensional functionally graded porous structure of multiple sheets Individual pieces, For stress tensor, For volume forces, For density, For displacement, For subfield symbols, For Dirichlet boundaries, For Neumann boundary, For gradient operators, The boundary displacement is defined on the Dirichlet boundary. The traction force specified at the Neumann boundary, This is the initial displacement. The initial velocity, The internal boundaries of the two sheets. Let be the normal vector of the first piece of material from its internal boundary outwards. Let be the normal vector of the second body from its inner boundary outwards. Let the stress tensor of the first sheet be... For the stress tensor of the second sheet, For the displacement of the first piece, For the displacement of the second piece, For the first The initial velocity of the sheet; In multi-sheet 3D-FGM porous structures, for small deformation and linear elastic problems, the strain tensor Expressed through geometric equations as follows: , Stress Tensor It can be derived from the physical equations as follows: ,in, for The elasticity matrix, , Stress Tensor and strain tensor It can be rewritten in vector form: , , That external normal vector on It can be rewritten in matrix form: , According to Nitsche's method, the trial solution space is defined. sum weight function space and find , making Satisfy the following formula: , , , , , in, It is a stable parameter that maintains the nonsingularity of the global stiffness matrix and ensures the stability of multi-piece solutions; Step 2: Use the modeling mesh as the initial analysis mesh; Step 3: Based on the elastic dynamics control equations, numerically solve the geometric configuration using geometric analysis such as PHT spline basis according to the analysis mesh to obtain the restored stress field; Step 4: Use the superconvergent restored basis error estimation algorithm to estimate the errors of the original stress field and the restored stress field to obtain the global relative error, and determine the subdomain with the maximum global relative error among multiple geometric configurations; Step 5: Use the Doffler marking strategy to mark the subdomain with the maximum error estimate; Step 6: Use the PHT spline local subdivision strategy to subdivide the marked subdomain to obtain the updated analysis mesh; Step 7: Repeat steps 3 to 6 until the global relative error is less than or equal to a preset threshold, then stop the iteration to obtain a numerical analysis template mesh that meets the error threshold; First, based on PHT-IGA, a superconvergent recovery basis error estimation algorithm and Dörfler labeling strategy are established to construct the PHT-AIGA algorithm. For the case where the PHT spline basis function has order P in each parameter direction, the algorithm is derived from the reference element... To element Mapping is defined ,element Each basis function on can be represented as A linear combination of Bernstein polynomials is given by the following equation: , in, It is an element Bessel coordinates of the basis functions It is the tensor product of Bernstein polynomials, in the reference interval The above definition can be expressed as: , , Its PHT spline basis functions are constructed in a hierarchical manner, for those having Level Layered T-grid, initial level The basis functions consist of B-spline basis functions defined on the tensor product grid. Through Bessel extraction, the basis functions are represented as linear combinations of Bernstein polynomials, thus obtaining the corresponding Bessel coordinates. The construction of the basis functions is a recursive process from one level to the next; the basis functions of subsequent levels are obtained by modifying the Bessel coordinates of the previous level. Let... Indicates the level as The base function set, hierarchy The set of basis functions is: , , , in, Indicates application to On the layer subdivision mesh Truncation operator for layer basis functions, This represents the newly introduced basis functions associated with the newly added vertices, referring to the new boundary vertices and new intersection vertices, rather than T-connection points; when considering the hierarchy... When subdividing certain cells, the de Casteljau algorithm is used to calculate the Bézier coordinates of the sub-cells. Then, the Bézier coordinates around the newly introduced vertices are set to zero through a truncation operation. Finally, the basis functions corresponding to the new vertices are derived using the Cox-de-Boor recursive formula. ,at last, The basis functions of the layer are and The union of these functions is iterated and repeated at successive levels until the final level is reached, ultimately generating a complete set of PHT spline basis functions. The geometric configuration in physical space is described by B-splines or NURBS, representing a mapping from the parameter space to the physical space. for: , in, It is the total number of control points in the geometric modeling mesh. The first one that provides geometric information One control point, Let represent the corresponding tensor product basis functions; then the PHT spline basis functions defined in physical space can be expressed as: , By discretization, the approximate displacement field can be represented by PHT spline basis functions: , in, It is the total number of control points in the discretized numerical analysis grid. and These are the PHT spline basis functions and the first... Unknown displacement of control points It can be represented in matrix form as follows: , The strain field can then be expressed as: , , The discrete equations for the multi-element free vibration problem based on the Nitsche method can be modified as follows: , in, It is a globally unknown displacement vector. , , and It can be represented as: , , , , An error estimation method based on superconvergence recovery, combined with the Dörfler labeling strategy, is employed to achieve adaptive mesh refinement. This process operates in two main stages: the first stage involves posterior error estimation at the small-body level using the recovered stress field; the second stage performs adaptive mesh refinement guided by the error distribution. To derive the recovered stress field, superconvergence points are used, where the numerical solution exhibits higher accuracy compared to the global convergence rate, especially for solutions satisfying certain conditions. of order and continuity The spline, the location of the superconvergence point in the reference domain. Please refer to Table 1 for the position above. Table 1
[0027] As shown in Table 1, the stress fields calculated by PHT-IGA at these specific locations... Sampling is performed, and the recovered stress field is constructed by approximating it with a higher-order polynomial over a local subdomain. , represented as ,exist Inside, set , Represents a set of conditions The non-overlapping subfields, where, Represents the number of subfields, operator It is expressed as follows: , in, It is supported in the subdomain superior A class Continuity The basis functions satisfy , , It is the variable to be solved. Subfield of The location of the superconvergence point; the stress recovery subdomain is defined. For an octet mesh, the first subdivision is performed globally for the initial coarse mesh, grouping every eight child cells belonging to the same parent into a subdomain. Subsequently, based on the Dörfler marking strategy, the marked subdomains are identified. and for each marked subdomain All eight units in the original subdomain were subdivided, resulting in each original subdomain being... It is divided into eight new subdomains, which are then used in the next step for stress recovery; subdomains Error in energy norm for: , Global error in its energy norm and global relative error for: , , Using the Dörfler labeling strategy, let... And select the subdomain with the largest error estimate such that: , Through iteration, the adaptive mesh refinement process continues until the global relative error is reduced to below the preset threshold, thus obtaining a numerical analysis template mesh that meets the error threshold.
[0028] In some embodiments, in step S103, a multi-objective optimization function is constructed with the optimization objectives of minimizing mass and maximizing the fundamental frequency of free vibration. The multi-objective optimization function is iteratively solved to obtain optimized design variables. The design variables at the control points of the design grid serve as the optimization basis. PHT-AIGA is used on the numerical analysis template grid to evaluate the population fitness during the multi-objective optimization process. An initial population of NSGA-III is constructed based on the design variables at the control points of the randomly initialized design grid. Each individual in the initial population has a unique material gradient distribution and porosity gradient distribution. PHT-AIGA is used on the numerical analysis template grid to evaluate the initial population fitness. The fitness of individuals in the initial population is evaluated, and superconvergent recovery basis error is estimated during each evaluation. When the global relative error is less than a set threshold, the calculation result is saved as the fitness of the individual in the population. When the global relative error is greater than the set threshold, the numerical analysis template grid is further refined until the global relative error is less than the threshold. The result is saved as the fitness of the current individual in the population. Offspring are generated through crossover and mutation operations of NSGA-III. Combining non-dominated sorting and reference point-based selection mechanisms, the process iteratively approaches the final Pareto front. Based on the Pareto front, optimized design variables are obtained, and its multi-objective optimization function is: , in, For designing the grid Material volume fraction at each control point For designing the grid Porosity at each control point For modeling the mesh Plate thickness at each control point It is the volume fraction of the material. The elastic modulus of the material, The mass density of the material, For dimensionless frequency, The geometric configuration characteristic length, For the initial uniform thickness, Let be the structural domain of the geometric configuration; the constraints of the multi-objective optimization function are: , in, The total integral of the material. The total porosity of the material; The established PHT-AIGA algorithm is embedded into the multi-objective optimization algorithm NSGA-III. Before multi-objective optimization, an initial multi-layer structure with uniform material and zero porosity is adaptively refined to generate a universal template mesh. This template mesh is used as the initial discretization mesh during the fitness evaluation of each individual in the optimization algorithm. Subsequently, considering the unique material and porosity gradient distribution characteristics of each individual, numerical analysis and global error estimation are performed. Then, a decision is made based on a predefined error tolerance. If the global relative error is lower than a predefined threshold, the template mesh is considered suitable for this individual, and the calculated performance meets the accuracy requirements, eliminating the need for further refinement. The performance evaluation results of each individual are stored. If the error exceeds the tolerance, the mesh is further refined for that specific individual until the specified accuracy is achieved. Multiple optimization objectives are set for the service requirements of the sample, and the range of values for design variables (material volume fraction and porosity) is determined. Corresponding constraints are set, and multi-objective optimization calculations are performed. Based on the Pareto front obtained from the multi-objective optimization calculations, representative elite solutions in the Pareto front are uniformly selected for result visualization. From the material and porosity distribution cloud maps, the characteristics of the optimization results under different design mesh resolutions, boundary conditions, and other factors are extracted. The influence of material and porosity distribution on the macroscopic performance of multiple 3D-FGM porous structures is analyzed.
[0029] Taking an L-shaped plate as an example, the geometric configuration of the L-shaped plate is established based on the NURBS method. For a schematic diagram of the mesh generation of the L-shaped plate, please refer to [link / reference needed]. Figure 2 ,like Figure 2 As shown in (a), the L-shaped plate consists of three pieces, namely Patch1, Patch2, and Patch3. The characteristic length of the L-shaped plate is... The initial uniform thickness of the plate is The L-shaped plate is composed of Al2O3 and Al. For its physical properties, please refer to Table 2. Table 2
[0030] The six sides of the L-shaped plate are completely fixed. The fundamental frequency of free vibration is characterized by a frequency infinity, and its expression is: , in, It is the first natural frequency of free vibration. It is a dimensionless frequency. , These are the mass density and elastic modulus of Al2O3, respectively. The L-shaped plate is initially adaptively meshed using PHT-AIGA, assuming it is composed of uniformly thick, non-porous Al2O3. The PHT spline order is set to [value missing]. Continuity set to The Dörfler flag parameter is set to Through iteration, the mesh is continuously subdivided adaptively until the global relative error falls below a set threshold, at which point the iteration stops and a template mesh is obtained. For the L-shaped plate, an independent hexahedron is created to generate the design mesh. A schematic diagram of the design mesh can be found in [reference needed]. Figure 3 ,like Figure 3 As shown, the design mesh completely encompasses the geometry of the L-shaped plate. Since the design mesh and the geometry are not completely coincident, some control points in the non-overlapping areas will not affect the optimization results. Therefore, while ensuring complete coverage of the target structure, the size of the design domain should be minimized as much as possible. By utilizing the symmetry of geometry and boundary conditions, a symmetrical design strategy is adopted to improve computational efficiency. Specifically, a diagonal mirror symmetry method is used in the in-plane direction, while an up-down symmetry method is used along the thickness direction. A multi-objective optimization problem is defined, aiming to maximize the fundamental frequency of the structure while minimizing its mass. Design variables include the ceramic volume fraction and porosity at the control points of the design mesh. To further improve structural performance, plate thickness design is also incorporated into the optimization. Unlike the material volume fraction and porosity, thickness design is limited to the planar direction and is directly implemented on the geometric modeling mesh. Based on the established PHT-AIGA-NSGA-III algorithm and the established multi-objective optimization problem, multi-objective optimization of multiple 3D-FGM porous structures is carried out. In the PHT-AIGA-NSGA-III algorithm, during the optimization process, a template mesh is used as the initial discretized numerical analysis mesh used by PHT-AIGA for evaluating the fitness of individuals in the population. If the accuracy of the calculation results meets the requirements, the results are saved and the next individual is calculated; otherwise, the mesh is further refined until the accuracy of the calculation results meets the requirements. The optimization results of three design mesh resolutions are compared, and their control point distributions are as follows: , and By comparing the Pareto frontier and visual optimization results, patterns were summarized, and the best optimized sample was obtained; the template mesh was obtained using PHT-AIGA, such as... Figure 2 As shown in (b), Figure 2In (b), the horizontal axis represents the degrees of freedom, and the vertical axis represents the relative error of the energy norm. The error convergence curves under adaptive mesh refinement and uniform mesh refinement are compared. Compared with uniform refinement, adaptive refinement achieves higher computational accuracy with fewer degrees of freedom and lower computational cost. Figure 2 (c)- Figure 2 (h) shows the relationship with Figure 2 (b) The mesh configuration in the corresponding adaptive refinement process shows that the refinement is mainly concentrated near the concave corner and the middle section of the two longer edges. The results show that the PHT-AIGA refinement method can achieve efficient mesh distribution, especially for complex geometries, and significantly reduces computational cost and time while maintaining high accuracy.
[0031] Please refer to the results of the Pareto front obtained by the L-shaped plate. Figure 4 ,like Figure 4 As shown, the Pareto front obtained from the multi-objective optimization of the L-shaped plate is presented, and the results of three design mesh resolutions are compared. The control point distribution is as follows: , and As the grid resolution increases, corresponding to the increase in the number of design variables, the Pareto front gradually moves towards the optimal direction (lower left corner), indicating that the structural performance is improved. However, for higher grid resolutions, a significant decrease in population diversity is observed. This is mainly attributed to the rapid expansion of the search space as the number of design variables increases. If the population size does not increase accordingly, the algorithm will have difficulty fully exploring the solution space and tends to converge toward a better objective value at the expense of solution diversity. from , and Please refer to the resulting graph showing the material distribution of representative elite solutions in the Pareto front of the designed mesh. Figure 5 , Figure 5 In the diagram, Representative eliteA, Representative eliteB, Representative eliteC, and Representative eliteD represent the material distribution of different representative elites. For the porosity distribution results of the representative elite solutions in the Pareto front of the designed mesh, please refer to [link / reference needed]. Figure 6 ,like Figure 5 and Figure 6 As shown, more design variables can better control the gradient distribution of materials and porosity; however, this increase also reduces population diversity, leading to elites becoming increasingly similar.
[0032] In summary, the optimization method for multi-layer multidimensional functionally graded materials (FJC) provided by this invention establishes modeling and design meshes based on B-splines or NURBS to construct multi-layer porous FJC structures. Each multi-layer porous FJC structure includes multiple control points and design variables defined at these control points, including material volume fraction and porosity. The constructed PHT-AIGA algorithm is used to adaptively solve and iteratively refine the geometric configuration of the multi-layer porous FJC structure, obtaining a numerical analysis template mesh that meets the error threshold. PHT-AIGA includes geometric analysis based on PHT splines, a superconvergent recovery basis error estimation algorithm, and a Doffler labeling strategy. A multi-objective optimization function is constructed with the optimization objectives of minimizing mass and maximizing the fundamental frequency of free vibration. The multi-objective optimization function is iteratively solved to obtain optimized design variables. The design variables at the control points of the design mesh serve as the optimization basis. Furthermore, PHT-AIGA is used on the numerical analysis template mesh to evaluate the population fitness during the multi-objective optimization process, improving the accuracy and efficiency of multi-objective optimization of multi-layer FJC.
[0033] To better implement the multi-layer multidimensional functional graded material optimization method in the embodiments of the present invention, based on the multi-layer multidimensional functional graded material optimization method, correspondingly, as follows: Figure 7 As shown, this embodiment of the invention also provides a multi-sheet multi-dimensional functional graded material optimization device, the multi-sheet multi-dimensional functional graded material optimization device 700, comprising: The porous structure construction module 701 is used to build modeling mesh and design mesh based on PHT splines or NURBS respectively to construct a multi-piece three-dimensional functional graded material porous structure. The multi-piece three-dimensional functional graded material porous structure includes multiple control points and design variables defined at the control points. The design variables include material volume fraction and porosity. The template mesh acquisition module 702 is used to adaptively solve and iteratively refine the geometric configuration of the porous structure of multiple three-dimensional functional graded materials using the constructed PHT-AIGA to obtain a numerical analysis template mesh that meets the error threshold. PHT-AIGA includes geometric analysis such as PHT spline basis, superconvergent recovery basis error estimation algorithm and Doveer marking strategy. The optimization module 703 is used to construct a multi-objective optimization function with the optimization objectives of minimizing mass and maximizing the fundamental frequency of free vibration. The multi-objective optimization function is iteratively solved to obtain the optimized design variables. The design variables at the control points of the design grid are used as the optimization basis. The population fitness during the multi-objective optimization process is evaluated on the numerical analysis template grid using PHT-AIGA.
[0034] like Figure 8As shown, the present invention also provides an electronic device 800, which can be a mobile terminal, desktop computer, laptop, handheld computer, server, or other computing device. The electronic device 800 includes a processor 801, a memory 802, and a display 803. Figure 8 Only some components of the electronic device 800 are shown, but it should be understood that it is not required to implement all the components shown, and more or fewer components may be implemented instead.
[0035] In some embodiments, memory 802 may be an internal storage unit of the electronic device 800, such as a hard disk or memory of the electronic device 800. In other embodiments, memory 802 may be an external storage device of the electronic device 800, such as a plug-in hard disk, smart media card (SMC), secure digital (SD) card, flash card, etc., equipped on the electronic device 800. Furthermore, memory 802 may include both internal and external storage units of the electronic device 800. Memory 802 is used to store application software and various types of data installed on the electronic device 800, such as program code installed on the electronic device 800. Memory 802 may also be used to temporarily store data that has been output or will be output. In one embodiment, memory 802 stores a multi-chip multi-dimensional functional graded material optimization program, which can be executed by processor 801 to implement the multi-chip multi-dimensional functional graded material optimization method of various embodiments of the present invention.
[0036] In some embodiments, processor 801 may be a central processing unit (CPU), microprocessor, or other data processing chip, used to run program code stored in memory 802 or process data, such as a multi-chip multidimensional functional gradient material optimization method.
[0037] In some embodiments, display 803 may be an LED display, a liquid crystal display, a touch-sensitive liquid crystal display, or an OLED (Organic Light-Emitting Diode) touchscreen. Display 803 is used to display identification information of the multi-dimensional functionally graded material optimization process and to display a visual user interface. Components 801-803 of electronic device 800 communicate with each other via a system bus.
[0038] In some embodiments, when the processor 801 executes the multi-sheet multidimensional functionally graded material optimization program in the memory 802, it implements each step of the multi-sheet multidimensional functionally graded material optimization method as described in the above embodiments. Since the multi-sheet multidimensional functionally graded material optimization method has been described in detail above, it will not be repeated here.
[0039] Accordingly, the present invention also provides a computer-readable storage medium for storing a computer-readable program or instruction, which, when executed by a processor, can implement the steps or functions in the multi-sheet multi-dimensional functionally graded material optimization method provided in the above-described method embodiments.
[0040] Those skilled in the art will understand that all or part of the processes of the methods described in the above embodiments can be implemented by a computer program instructing related hardware, and the program can be stored in a computer-readable storage medium. The computer-readable storage medium may be a disk, optical disk, read-only memory, or random access memory, etc.
[0041] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.
Claims
1. A method for optimizing multi-layered multidimensional functionally graded materials, characterized in that, include: Modeling meshes and design meshes are established based on B-splines or NURBS respectively to construct multi-sheet three-dimensional functional graded material porous structures. The multi-sheet three-dimensional functional graded material porous structures include multiple control points and design variables defined at the control points. The design variables include material volume fraction and porosity. The constructed PHT-AIGA is used to adaptively solve and iteratively refine the geometric configuration of porous structures of multi-layer three-dimensional functional graded materials to obtain a numerical analysis template mesh that meets the error threshold. The PHT-AIGA includes geometric analysis such as PHT spline basis, superconvergent recovery basis error estimation algorithm and Doveer marking strategy. A multi-objective optimization function is constructed with the objectives of minimizing mass and maximizing the fundamental frequency of free vibration. The multi-objective optimization function is iteratively solved to obtain the optimized design variables. The design variables at the control points of the design grid are used as the optimization basis. The population fitness during the multi-objective optimization process is evaluated on the numerical analysis template grid using the PHT-AIGA.
2. The method for optimizing multi-layer multidimensional functionally graded materials according to claim 1, characterized in that, The three-dimensional functionally graded material is composed of metallic and ceramic materials; the construction of a multi-layer porous structure of the three-dimensional functionally graded material based on B-splines or NURBS to establish modeling meshes and design meshes respectively includes: Modeling meshes are built based on B-splines or NURBS to construct the geometric configurations of multiple three-dimensional functionally graded materials; A design mesh is established based on B-splines or NURBS to construct a monolithic hexahedral design domain, wherein the monolithic hexahedral design domain completely covers the geometric configuration; Interpolate the design variables at the control points in the design grid based on B-splines or NURBS to obtain the value of the design variables at any point in the hexahedral design domain; By mapping, the values of design variables in the hexahedral design domain established by the design mesh are transferred to the geometric configuration positions established by the modeling mesh under the corresponding coordinates, so as to construct a multi-layer three-dimensional functional graded material porous structure.
3. The method for optimizing multi-layer multidimensional functionally graded materials according to claim 2, characterized in that, The constructed PHT-AIGA is used to adaptively solve and iteratively refine the geometry of porous structures of multiple three-dimensional functionally graded materials to obtain a numerical analysis template mesh that meets the error threshold, including: Step 1: Establish the elastic dynamics governing equations for porous structures of multi-layer three-dimensional functionally graded materials based on three-dimensional elasticity theory and Nitsche's method; Step 2: Use the modeling mesh as the initial analysis mesh; Step 3: Based on the aforementioned elastic dynamics control equations, the geometric configuration is numerically solved using geometric analysis methods such as PHT spline basis according to the analysis mesh to obtain the restored stress field; Step 4: Use the superconvergent recovery basis error estimation algorithm to estimate the error of the original stress field and the recovered stress field, obtain the global relative error, and determine the subdomain with the maximum global relative error in the multi-piece geometric configuration; Step 5: Use the Doffler labeling strategy to label the subdomain of the maximum error estimate; Step 6: Based on the PHT spline local subdivision strategy, subdivide the marked subdomains into a mesh to obtain the updated analysis mesh; Step 7: Repeat steps 3 to 6 until the global relative error is less than or equal to a preset threshold, then stop the iteration to obtain a numerical analysis template mesh that meets the error threshold.
4. The optimization method for multi-layer multidimensional functionally graded materials according to claim 3, characterized in that, The governing equations of elastic dynamics are: ， ， ， ， ， ， ， in, For the first three-dimensional functionally graded porous structure of multiple sheets Individual pieces, For stress tensor, For volume forces, For density, For displacement, For subfield symbols, For Dirichlet boundaries, For Neumann boundary, For gradient operators, The boundary displacement of the subdomain on the Dirichlet boundary. The traction force specified by the Neumann boundary for the subdomain. This is the initial displacement. The initial velocity, For the internal boundaries of the two sheets, Let be the normal vector of the first piece of material from its internal boundary outwards. Let be the normal vector of the second body from its inner boundary outwards. Let the stress tensor of the first sheet be... For the stress tensor of the second sheet, For the displacement of the first piece, For the displacement of the second piece, For the first The initial velocity of the sheet.
5. The method for optimizing multi-layer multidimensional functionally graded materials according to claim 3, characterized in that, The process involves constructing a multi-objective optimization function with the objectives of minimizing mass and maximizing the fundamental frequency of free vibration. This multi-objective optimization function is then iteratively solved to obtain the optimized design variables, including: An initial population of NSGA-III is constructed based on the design variables at the control points of the design grid, where each individual in the population has a unique material gradient distribution and porosity gradient distribution. The fitness of individuals in the initial population is evaluated using PHT-AIGA on the numerical analysis template grid. During each evaluation, superconvergent recovery basis error estimation is performed. When the global relative error is less than a set threshold, the calculation result is saved as the fitness of the individual in the population. When the global relative error is greater than the set threshold, the numerical analysis template grid is further refined until the global relative error is less than the threshold. The result is saved as the fitness of the current individual in the population. Offspring are generated through crossover and mutation operations using NSGA-III. Combining non-dominated sorting and a reference point-based selection mechanism, the system iteratively approximates the final Pareto front, and optimized design variables are obtained based on the Pareto front.
6. The method for optimizing multi-layer multidimensional functionally graded materials according to claim 5, characterized in that, The multi-objective optimization function is: ， in, For designing the grid Material volume fraction at each control point For designing the grid Porosity at each control point For modeling the mesh Plate thickness at each control point It is the volume fraction of the material. The elastic modulus of the material, The mass density of the material, For dimensionless frequency, The characteristic length of the geometric configuration, For the initial uniform thickness, It is a geometric configuration structural domain.
7. The method for optimizing multi-layer multidimensional functionally graded materials according to claim 6, characterized in that, The constraints of the multi-objective optimization function are: ， in, The total integral of the material. The total porosity of the material.
8. A device for optimizing multi-sheet multi-dimensional functionally graded materials, characterized in that, include: A porous structure construction module is used to establish a modeling mesh and a design mesh based on B-splines or NURBS, respectively, to construct a multi-sheet three-dimensional functional graded material porous structure. The multi-sheet three-dimensional functional graded material porous structure includes multiple control points and design variables defined at the control points. The design variables include material volume fraction and porosity. The template mesh acquisition module is used to adaptively solve and iteratively refine the geometric configuration of the porous structure of multiple three-dimensional functional graded materials using the constructed PHT-AIGA to obtain a numerical analysis template mesh that meets the error threshold. The PHT-AIGA includes geometric analysis such as PHT spline basis, superconvergent recovery basis error estimation algorithm and Doveer marking strategy. The optimization module is used to construct a multi-objective optimization function with the objectives of minimizing mass and maximizing the fundamental frequency of free vibration. The multi-objective optimization function is iteratively solved to obtain the optimized design variables. The design variables at the control points of the design grid are used as the optimization basis, and the population fitness during the multi-objective optimization process is evaluated on the numerical analysis template grid using the PHT-AIGA.
9. An electronic device, characterized in that, Including memory and processor; The memory stores a computer-readable program that can be executed by the processor; When the processor executes the computer-readable program, it implements the steps in the multi-sheet multidimensional functionally graded material optimization method as described in any one of claims 1-7.
10. A computer-readable storage medium, characterized in that, Used to store computer-readable programs or instructions, which, when executed by a processor, can implement the steps in the multi-sheet multidimensional functionally graded material optimization method according to any one of claims 1-7.